Minimum Mean Square Error (Wiener) Filtering

The Wiener Filter is a classical approach in image restoration, aiming to reduce the degradation caused by factors such as noise and blurring in an image. The method is grounded in the assumption that both the image and the noise are random variables, and the objective is to estimate an uncorrupted image $\hat{f}$ such that the mean square error (MSE) between the original image $f$ and the estimate $\hat{f}$ is minimized.

Table of Contents

Mathematical Foundation

The error measure is formulated as follows:

e^2 = E\{(f – \hat{f})^2\}

Where:

$E \{ \cdot \}$ represents the expected value of the argument.
$f$ is the original image.
$\hat{f}$ is the estimate of the uncorrupted image.

The Wiener filter leverages this error criterion to optimize the restoration process by incorporating both the degradation function and statistical properties of the noise. The minimum mean square error (MMSE) can be expressed in the frequency domain as:

\hat{F}(u,v) = \frac{H^*(u,v) S_f(u,v)}{|H(u,v)|^2 S_f(u,v) + S_n(u,v)} G(u,v)

Here:

$H(u,v)$ is the degradation function.
$H^*(u,v)$ is the complex conjugate of the degradation function.
$S_f(u,v)$ is the power spectrum of the uncorrupted image.
$S_n(u,v)$ is the power spectrum of the noise.
$G(u,v)$ is the degraded image in the frequency domain.

This equation demonstrates how the Wiener filter combines the image’s power spectrum with that of the noise to perform optimal filtering.

Signal-to-Noise Ratio (SNR)

One of the most important metrics in evaluating the performance of image restoration techniques, such as the Wiener filter, is the Signal-to-Noise Ratio (SNR). It quantifies the level of the signal (the original image information) compared to the noise level.

The SNR in the frequency domain is defined as:

SNR = \frac{\sum_{u=0}^{M-1} \sum_{v=0}^{N-1} |F(u,v)|^2}{\sum_{u=0}^{M-1} \sum_{v=0}^{N-1} |N(u,v)|^2}

Where:

$F(u,v)$ is the Fourier transform of the original (undegraded) image.
$N(u,v)$ is the Fourier transform of the noise.

This ratio helps in understanding the information content in the image relative to the noise. A higher SNR indicates a clearer image with less noise, while a lower SNR suggests a noisier image.

In the spatial domain, the SNR can be expressed as:

SNR = \frac{\sum_{x=0}^{M-1} \sum_{y=0}^{N-1} f(x,y)^2}{\sum_{x=0}^{M-1} \sum_{y=0}^{N-1} \left( f(x,y) – \hat{f}(x,y) \right)^2}

Where:

$f(x,y)$ is the original image.
$\hat{f}(x,y)$ is the restored image.

Mean Square Error (MSE)

Another commonly used metric for image restoration performance is the Mean Square Error (MSE), which measures the average squared difference between the original image $f(x,y)$ and the restored image $\hat{f}(x,y)$ :

MSE = \frac{1}{MN} \sum_{x=0}^{M-1} \sum_{y=0}^{N-1} \left( f(x,y) – \hat{f}(x,y) \right)^2

A smaller MSE indicates a better restoration result, meaning that the restored image is closer to the original image.

Wiener Filter for White Noise

When dealing with spectrally white noise (where the power spectrum of the noise is constant), the Wiener filter reduces to a simpler form, as the power spectrum of the noise $|N(u,v)|^2$ becomes a constant. In such cases, when the power spectra of the undegraded image are unknown, the Wiener filter is approximated by introducing a constant $K$ to all terms of the degradation function:

\hat{F}(u,v) = \frac{1}{H(u,v)} \frac{|H(u,v)|^2}{|H(u,v)|^2 + K} G(u,v)

Here, $K$ is chosen interactively based on the visual quality of the result.

**Fig 5.28:** Comparison of inverse and Wiener filtering. (a) The result of full inverse filtering is shown in Fig. 5.25(b). (b) Radially limited inverse filter result. (c) Wiener filter result.

**Fig 5.29:** (a) 8-bit image corrupted by motion blur and additive noise. (b) Result of inverse filtering. (c) Result of Wiener filtering. (d)–(f) Same sequence, but with noise variance one order of magnitude less. (g)(i) Same sequence, but noise variance reduced by five orders of magnitude from (a). Note in (h) how the deblurred image is quite visible through a “curtain” of noise.

Examples and Applications

In Figure 5.28, the advantage of Wiener filtering over direct inverse filtering is evident. The following three images are compared:

The fully inverse-filtered result from Figure 5.27(a).
The radially limited inverse filter result from Figure 5.27(c).
The Wiener filter result.

The comparison reveals that Wiener filtering delivers an image much closer to the original, effectively reducing noise while preserving the structure of the image.

In Figure 5.29, the effect of additive Gaussian noise and motion blur on an image is illustrated. The Wiener filter shows a significant improvement over direct inverse filtering, especially when the noise variance is reduced by multiple orders of magnitude. The text in the images becomes more readable as the Wiener filter removes much of the noise and blur.

Conclusion

The Wiener Filter is a robust technique in image restoration that offers a balance between noise reduction and signal preservation. By minimizing the mean square error, the Wiener filter ensures that the restored image closely approximates the original uncorrupted image. While not perfect, it provides significant improvements over simpler inverse filtering techniques, especially in scenarios where noise characteristics are known or can be approximated.

References

Gonzalez, R. C., & Woods, R. E. (2018). Digital Image Processing (4th ed.). Pearson.
Wiener, N. (1949). Extrapolation, Interpolation, and Smoothing of Stationary Time Series. MIT Press.
Pratt, W. K. (2007). Digital Image Processing: PIKS Scientific Inside (4th ed.). Wiley-Interscience.
Jain, A. K. (1989). Fundamentals of Digital Image Processing. Prentice-Hall.

Mathematical Foundation

Signal-to-Noise Ratio (SNR)

Mean Square Error (MSE)

Wiener Filter for White Noise

Examples and Applications

Conclusion

References

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